Charged black hole - static electric field lines

[dllahr]

Where do the static electric field lines appear to originate from a charged black hole, non rotating, Reissner–Nordström metric?

I've had a number of qualified physicists say they appear to come from the center of the black hole, but people on these forums have said that doesn't make sense. They've also gone so far as to claim the singularity is not at the center of the black hole ... ?

 

[Nugatory]

There are two things going on here.

First, you can indeed find qualified physicists who will say that the charge lines "appear to come from the center of the black hole". That's a simplified picture appropriate for laypeople who won't be helped by a digression into why the phrase "center of the black hole" doesn't mean what it sounds like; and there's nothing wrong with it as long as you understand the limitations of that explanation and don't read too much into it. However, in an I-level thread we expect the discussion to go beyond that level.

Second, "appear to come from the center of the black hole" and "the charge...is located at the singularity" are not equivalent statements. The first is a statement about what the charge lines look like outside the black hole, whereas the second is a statement about the location of the charge (and a bogus one at that, because nothing can be "located" at the center).

 

[PeterDonis]

the second is a statement about the location of the charge (and a bogus one at that, because nothing can be "located" at the center).

There are actually several points involved here:

In Schwarzschild spacetime, the locus $r = 0$ is a spacelike line, not a timelike line, so it isn't a "location" at all--it's a moment in time, not a place in space.

In Reissner-Nordstrom spacetime, the locus $r = 0$ is timelike, but it is two timelike lines, not one, and they're inside the inner horizon, so they are not spacelike separated from any event outside the horizon. So describing this locus as "a place at the center of the hole" doesn't seem correct. (And for good measure, it's actually an infinite number of pairs of timelike lines, if you look at the maximal extension of R-N spacetime.)

Finally, R-N spacetime is an electrovacuum solution, which means there is zero charge density everywhere, so asking "where" the charge is located is meaningless, since there isn't any.

 

[dllahr]

There are two things going on here.

First, you can indeed find qualified physicists who will say that the charge lines "appear to come from the center of the black hole". That's a simplified picture appropriate for laypeople who won't be helped by a digression into why the phrase "center of the black hole" doesn't mean what it sounds like; and there's nothing wrong with it as long as you understand the limitations of that explanation and don't read too much into it. However, in an I-level thread we expect the discussion to go beyond that level.

Center of the black hole is a pretty obvious concept, I'm not sure why you're trying to obfuscate it. Regardless of the level of the discussion, it serves as a good starting point. I'm not sure exactly what l-level means; I'd be happy to move the discussion elsewhere if that's going to be a big hang up.

There are actually several points involved here:

In Schwarzschild spacetime, the locus $r = 0$ is a spacelike line, not a timelike line, so it isn't a "location" at all--it's a moment in time, not a place in space.

In Reissner-Nordstrom spacetime, the locus $r = 0$ is timelike, but it is two timelike lines, not one, and they're inside the inner horizon, so they are not spacelike separated from any event outside the horizon. So describing this locus as "a place at the center of the hole" doesn't seem correct. (And for good measure, it's actually an infinite number of pairs of timelike lines, if you look at the maximal extension of R-N spacetime.)

Finally, R-N spacetime is an electrovacuum solution, which means there is zero charge density everywhere, so asking "where" the charge is located is meaningless, since there isn't any.

Again, this is unnecessarily complicated, and misses the point entirely. From outside the outer event horizon, and a reasonable distance, we don't need to worry about if "the locus r = 0 is spacelike or timelike". It's ridiculous to say there isn't any charge - the black hole is charged. There are static electric field lines emanating from the black hole. According to the Reissner-Nordstrum metric you would measure static electric field lines, and it would appear as if they are originating from the center of the black hole. Good old vector calculus - Gauss's Law - comes in handy here.

I think the major problem here is that you're not trying to imagine running actual experiments near the black hole. That's what science is, right? You have ideas, then you test them with experiments, then repeat. When I propose a simple experiment, using well established measurement techniques, you shouldn't say "You can't do that!". You should consider what the readout from the measurement device would be.

How about some actual physics?
https://journals.aps.org/prd/abstract/10.1103/PhysRevD.33.915

From the membrane viewpoint of black holes, the horizon

which is viewed as a 2-dimensional membrane that resides in 3-dimensional space and evolves in response to driving forces from the external universe. This membrane, following ideas of Damour and Znajek, is regarded as made from a 2-dimensional viscous fluid that is electrically charged and electrically conducting and has finite entropy and temperature, but cannot conduct heat.​

 

[berkeman]

I'm not sure exactly what l-level means;

https://www.physicsforums.com/threads/what-do-the-thread-prefixes-mean-a-i-b.933573/

 

[PeterDonis]

Center of the black hole is a pretty obvious concept

No, it isn't. That's why we're going to the trouble to correct you when you use it according to your "obvious" intuition, since that intuition is wrong when applied to a black hole.

It's ridiculous to say there isn't any charge - the black hole is charged.

Then please show me where, in the Reissner-Nordstrom metric, there is nonzero charge density. If you can do so, it will be quite a trick, since the R-N metric is, as I said, an electrovacuum solution, meaning that there is zero stress-energy (except for the EM field) and zero charge density everywhere.

There are static electric field lines emanating from the black hole.

Outside the horizon, yes, this is a reasonable view. However, inside the horizon, the spacetime is no longer static, and the concept of "static field lines" no longer makes sense.

it would appear as if they are originating from the center of the black hole

Nope, sorry, here you are applying intuitions where they are not valid.

How about some actual physics?

What does this have to do with your claims about the "center" of a R-N black hole? Nothing, as far as I can see, since it only talks about the horizon and the region outside it.

 

[dllahr]

No, it isn't. That's why we're going to the trouble to correct you when you use it according to your "obvious" intuition, since that intuition is wrong when applied to a black hole.
Then please show me where, in the Reissner-Nordstrom metric, there is nonzero charge density. If you can do so, it will be quite a trick, since the R-N metric is, as I said, an electrovacuum solution, meaning that there is zero stress-energy and zero charge density everywhere.
Outside the horizon, yes, this is a reasonable view. However, inside the horizon, the spacetime is no longer static, and the concept of "static field lines" no longer makes sense.
Nope, sorry, here you are applying intuitions where they are not valid.
What does this have to do with your claims about the "center" of a R-N black hole? Nothing, as far as I can see, since it only talks about the horizon and the region outside it.

Yes it is. The Reissner-Nordstrum metric has a spherically symmetric horizon. Bingo, center of that sphere. Easy.

All of my questions - in this thread and the other - have focused very specifically on making measurements outside the horizon. The horizon that is spherically symmetric. And hence has a well-defined center.

The black hole is charged. I've never said "Where is the charge?" That's something you appear to be hung up on. I've always very clearly asked where does it appear to be, based on measurements of the electrostatic field.

 

[PeterDonis]

The Reissner-Nordstrum metric has a spherically symmetric horizon. Bingo, center of that sphere. Easy.

Sorry, you're still using intuitions where they don't apply. The fact that you have a 2-sphere does not imply that there is a center in all cases. It does in flat Minkowski spacetime, and in many other spacetimes, but it doesn't in a black hole spacetime.

All of my questions - in this thread and the other - have focused very specifically on making measurements outside the horizon.

If that is true, then why do you care where the "center" is, or even whether there is one? Why do you care where the field lines you measure outside the horizon "appear to come from", or even whether that concept makes sense for a black hole?

You have said what's important is experiments. What experiments would you expect to come out differently if there is a "center", vs. if there isn't?

I've had a number of qualified physicists say they appear to come from the center of the black hole

Arguments from authority carry no weight. However, you did give one piece of math: the 4-potential $A_\alpha = \left( Q / r, 0, 0, 0 \right)$. And if you want to define "appear to come from the center of the black hole" as "the 4-potential has this form", then that's just a definition of terms which you are free to make (although it would be nice to show us some actual peer-reviewed papers that define terms this way, so we know it isn't just your own personal definition). But definitions of terms do not affect the physics, and disagreement about terms does not necessarily reflect disagreement about physics.

If you agree that the physics I have given--that the R-N metric is an electrovacuum solution, that it has zero stress-energy (other than the EM field) and zero charge density everywhere, and that it is not static inside the horizon, and so the concept of "static field lines" makes no sense there--is correct, then we are in agreement and we just are making different choices of terminology. If you don't agree that the physics I have given is correct, then you're going to have to explain why not.

 

[dllahr]

Sorry, you're still using intuitions where they don't apply. The fact that you have a 2-sphere does not imply that there is a center in all cases. It does in flat Minkowski spacetime, and in many other spacetimes, but it doesn't in a black hole spacetime.

Nope; apparently you didn't read my whole post, and just fired that off. Or my other posts. I've posited repeatedly that I'm making measurements from outside the outer event horizon, so sphere's do have a center. Why are you arguing against things I'm not saying?

If that is true, then why do you care where the "center" is, or even whether there is one? Why do you care where the field lines you measure outside the horizon "appear to come from", or even whether that concept makes sense for a black hole?

You have said what's important is experiments. What experiments would you expect to come out differently if there is a "center", vs. if there isn't?

I never said I cared where the center is; I cared where it appeared that the charge was located based on electrostatic field lines. Again, maybe you should try reading what I wrote.

Arguments from authority carry no weight. However, you did give one piece of math: the 4-potential $A_\alpha = \left( Q / r, 0, 0, 0 \right)$. And if you want to define "appear to come from the center of the black hole" as "the 4-potential has this form", then that's just a definition of terms which you are free to make (although it would be nice to show us some actual peer-reviewed papers that define terms this way, so we know it isn't just your own personal definition). But definitions of terms do not affect the physics, and disagreement about terms does not necessarily reflect disagreement about physics.

If you agree that the physics I have given--that the R-N metric is an electrovacuum solution, that it has zero stress-energy (other than the EM field) and zero charge density everywhere, and that it is not static inside the horizon, and so the concept of "static field lines" makes no sense there--is correct, then we are in agreement and we just are making different choices of terminology. If you don't agree that the physics I have given is correct, then you're going to have to explain why not.

I think the only reason we're not arguing is that you've finally decided to read my questions instead of "arguing" about things I wasn't asking or claiming.

 

[PeterDonis]

I've posited repeatedly that I'm making measurements from outside the outer event horizon, so sphere's do have a center.

Sorry, this is still not correct. Measurements outside the horizon can tell you that the horizon is a 2-sphere, but they cannot tell you that the 2-sphere has a center. To know that you would have to make measurements inside the 2-sphere, i.e., inside the horizon.

(At least, the above is true with the standard meaning of the term "center" in geometry. But at this point I have no idea what you mean by the word. See below.)

I never said I cared where the center is; I cared where it appeared that the charge was located based on electrostatic field lines.

So "where it appears the charge was located" is not the same as "where the center is"? You have been using those two terms synonymously up to now.

Can you give explicit definitions for what you think those two terms mean? That might help since at this point I have no idea what you are actually trying to say.

 

[PeterDonis]

Why are you arguing against things I'm not saying?

I'm not. I'm pointing out that the things you are saying are either wrong or don't make sense. At least not with the meanings of words that I'm familiar with.

 

[pervect]

Well, Gauss's law works just fine in special relativity, and in General Relativity it's quite similar, due to the way E&M can be modeled by a differential form, which makes the "lines of force" concpet work pretty much the same in GR as it does in SR.

Because the Schwarzschild is a vacuum solution, if a 3-sphere doesn't enclose the black hole, the charge is zero. Only if it includes the singularity is the enclosed charge (by the appropriate integration of the lines of force) nonzero.

It's clear to me that a sphere outside the event horizon encloses a net charge. Geometrically the number of field lines emenating from a black hole is constant, so the density of field lines per unit area, which is proportioanl to the field, goes down as the area of the enclosing 3-sphere goes up, keeping the product constant.

I'd have to think hard about what happens inside the BH, though.

It's also clear that a sphere enclosing only the vacuum region of space-time and none of the singularity will enclose zero charge.

 

[PeterDonis]

Gauss's law works just fine in special relativity, and in General Relativity it's quite similar

That's true, and nobody has disputed that in this discussion.

Because the Schwarzschild is a vacuum solution, if a 3-sphere doesn't enclose the black hole, the charge is zero. Only if it includes the singularity is the enclosed charge (by the appropriate integration of the lines of force) nonzero.

Note that, since we are talking about a charged black hole, we are using the Reissner-Nordstrom solution, not Schwarzschild. The Schwarzschild solution describes an electrically neutral hole.

It's clear to me that a sphere outside the event horizon encloses a net charge.

In the sense that Gauss's Law gives a nonzero answer, yes, and it's easy to show that that charge is $Q$, the number that appears in the $Q^2 / r^2$ term in the $g_{tt}$ and $g_{rr}$ metric coefficients. Nobody is disputing that either.

It's also clear that a sphere enclosing only the vacuum region of space-time and none of the singularity will enclose zero charge.

Here is where it gets sticky. What does a given 2-sphere "enclose"? The usual sense of this term is that we have some spacelike 3-surface containing the 2-sphere, and this surface can be foliated by an infinite family of 2-spheres, including the one we're looking at, whose surface areas increase smoothly from zero, through the area of our chosen 2-sphere, and on up to infinity. Then we say our chosen 2-sphere encloses the 3-ball comprised of all the 2-spheres with surface area less than our chosen one.

The problem in a black hole spacetime is that the 3-surfaces of constant time for a static observer outside the horizon do not have the property I just described. Instead they are foliated by an infinite family of 2-spheres, whose surface areas start out at infinity (at spatial infinity on the same side of the horizon as the static observer), decrease down to the surface area of the horizon, and then increase back up to infinity on the other side. (Note that we are talking about the maximally extended geometry here.) The singularity, and indeed any 2-sphere with surface area smaller than that of the horizon, does not occur anywhere in this 3-surface.

In the Schwarzschild case, we can find other spacelike 3-surfaces that include the static observer and also include the singularity and 2-spheres inside the horizon (for example, a surface of constant Painleve coordinate time). But in that case, the singularity itself is a spacelike surface, so it is a moment of time, not a place in space, and it doesn't make sense to say that any 2-sphere "encloses" it.

In the Reissner-Nordstrom case, the singularities (I use the plural for reasons explained in an earlier post) are inside the inner horizon, and there are no spacelike surfaces whatsoever that cover both the region inside the inner horizon and the region outside the outer horizon. So even though the singularities are timelike, and therefore can be thought of as "places in space", they are not places in any "space" that includes the region outside the outer horizon, and it doesn't make sense to say that any 2-sphere outside the outer horizon "encloses" them.

Such language might possibly make sense for an observer that was inside the inner horizon, doing the Gauss's Law integral over a 2-sphere containing him. However, in that case it's no longer clear which of the two singularities his 2-sphere can be said to "enclose"; the spacelike 3-surfaces in this region are foliated by 2-spheres whose surface area starts at zero (at one singularity), increases to the surface area of the inner horizon, and then decreases back to zero again (at the other singularity). So even in this case it's not clear that the usual physical interpretation of the Gauss's Law integral is valid.

 

[dllahr]

Sorry, this is still not correct. Measurements outside the horizon can tell you that the horizon is a 2-sphere, but they cannot tell you that the 2-sphere has a center. To know that you would have to make measurements inside the 2-sphere, i.e., inside the horizon.

(At least, the above is true with the standard meaning of the term "center" in geometry. But at this point I have no idea what you mean by the word. See below.)
So "where it appears the charge was located" is not the same as "where the center is"? You have been using those two terms synonymously up to now.

Can you give explicit definitions for what you think those two terms mean? That might help since at this point I have no idea what you are actually trying to say.

Nope. I asked if the charge was located at the apparent center. I asked where the charge was located. Really simple if you stop arguing at straw men.

 

[Nugatory]

This thread closed